These are some notes.
There is a local term for each edge that is the indicator function on whether the assignments to the two vertices are different.
We consider the following example.
Even before we talk about quantum we look at the most basic unit of information. It can take one of two values: typically, 0 or 1.
We can quantumize a bit into qubit. In short, it is a superposition of 0 and 1. Lets break that down.
We have our bits. We turn them into ...
We turn them into the so called computational basis vectors.
Which can be understood as vectors. Indeed we will use the bra-ket notation, where a ket, |v>, is a vectors.
A general quantum state takes the form of a linear combination. Here, we think of a superposition as represented by a linear combination.
that is a unit vector. Note the the <\psi| is called a bra, which is the same as a row vectors, or the conjugate transpose of the ket version (or, rather, a dual vector). Putting a bra and a ket together like this gives an inner product.
This gives us the following definition for a qubit.
Lets break that down.
classically, we understand multiple bits with the cartesian product. That is, the cartesian product gives us the set of all length 2 bit strings. As with the single qubit case, we assign to each string a computation basis vectors...
As with the single qubit case, we assign to each string a computation basis vectors. We can decompose each computation basis vector into the tensor product of the individual computation basis vectors.
All together, a state on two qubits is just a linear combination of the computation basis vectors.
lets break this down and look at a special case.
If I change the scalars to be the following, we get something interesting.
Lets expand it out.
re-group.
re-group again. Here we notices something interesting. This is just a tensor product of two single qubit states.
So, in a sense, entanglement is a result of superpositions over multiple qubits/states.
Recall where we left off with classical optimization.
we just put the value into the diagonal entry indexed by the strings.
Indeed, this means the value is just quadratic form of the computation basis vector. Moreover, we have turned this optimization problem into an eigenvalue problem.
Here we generalize the notion on the left, that observables are represented by diagonal matrices, with the full quantum version that they need only be diagonalizable or rather self-adjoint.
We call this the Hamiltonian and the H_alpha's the local hamiltonians.
the quantity of most importance is teh quadratic form, which we call the energy. Instead of taking the quadratic form over only computation basis vectors we consider arbitrary states.
the quantity of most importance is teh quadratic form, which we call the energy. Instead of taking the quadratic form over only computation basis vectors we consider arbitrary states.
Lets dissect this a little.
Lets dissect this a little. Recall Max-cut... We use this notation to denote the outer product.
In QMC we not only want the state to be in the "different" subspace but we also want there to be this quantum anti-correlation between the two ways of being different. This is called the antisymmetric subspace and is of extreme important in the study of quantum information.